Spectroscopy and the Particle in a Box

Transcription

1 Spectroscopy and the Particle in a Box Introduction The majority of colors that we see result from transitions between electronic states that occur as a result of selective photon absorption. For a molecule to absorb a photon, the energy of the impinging photon must match the energy difference between the initial state and some excited state of the molecule. We can describe this concept using the equation E photon = hυ = ΔE molecule = E upper state - E lower state (1) in which E represents the energy of the photon or molecule being studied, h is Planck's constant, and υ is the frequency. To predict the color of a specific molecule from fundamental physical chemistry principles, one must know the array of possible molecular energy levels (quantized rotational, vibrational, and electronic energy levels). Molecules of a colored object absorb visible-light photons when they are excited from their lowest-energy electronic state (called the ground state ) to a higher-energy electronic state (called an excited state ). In principle, the various electronic states of an atom or molecule may be calculated quantummechanically. In fact, quantum mechanics can be used to predict the allowed set of energy levels for an atom or molecule. For larger molecules, determining these energy levels requires making approximations and it can be computationally intensive (For simpler molecules, free software is available that would allow you to carry out such calculations on your home computer). With a carefully chosen set of molecules, however, we can study some of the principles of quantum mechanics in the general chemistry laboratory. Some ultraviolet (UV) light and visible light-absorbing molecules are members of a special group for which the simple "particle in a box" quantum-mechanical model applies nicely. This model can be used to predict the energy levels of electrons responsible for UV or visible wavelength transitions if we are willing to make some assumptions. Imagine that a particle of mass m (in this case, the particle is an electron) travels in one dimension (x) between two walls separated by a distance L. Then assume that the potential 1

2 energy between these walls (i.e., from 0 x L) is constant, while the potential energy jumps to infinity at the walls. This assumption allows us to draw a simple potential energy diagram like that shown on page one. Solving the Schrödinger equation for this simple one-dimensional particle in a box system yields the following allowed energies: E n = n2 h 2 8mL 2 n = 1,2,3,... where h is Planck's constant, m is the mass of the particle, and L is the length of the onedimensional box. Note that the different allowed energies are labeled by the quantum number n, which can only take on integer values. If the energy of a particle in a box is measured, these are the only results that will be found no other energies are possible results according to quantum mechanics. These energies can be qualitatively understood by considering the wave-particle duality in quantum mechanics, wherein objects we normally think of as particles in some ways behave as waves and vice-versa. This means that an electron in a box must be described as a wave in quantum mechanics with wavelength λ=h/p. This wave is related to the probability of finding the electron in fact the electron probability distribution of positions is the square of the wavefunction that describes the particle and hence the wave must go to zero outside of the box where the potential energy is infinitely high. That is, the wave describing the electron must fit neatly in the box so that it goes to zero at the edges. This is clearly the case when the length of the box, L, is equal to λ/2, λ, 3λ/2, 2λ, 5λ/2,... Combining this with the expression for λ above and using E=mv 2 /2 gives the allowed energies. Note that this somewhat hand-waving explanation also shows how the energy levels and their spacings depend on L and m. As L becomes very large the energies get closer and closer together, eventually becoming continuous (no longer quantum ); this is due to L being much larger than wavelength λ for a particle with a typical energy. Similarly, increasing the mass has the same qualitative effect as making the box larger, which is why you (a particle ) do not notice quantum effects when you sit in a room (a box ), even though your motion is fundamentally described by quantum mechanics. Stated another way, discrete energy level spacing is observed for very low-mass particles confined to small quarters (in this case, an electron within an atom or molecule that gives a small value for ml 2 ). For the molecules considered in this experiment, the electronic energy level spacing corresponds to the energy of a visible photon. Specifically, there must be high-energy valence electrons capable of traveling "freely" over the length of the molecule, L. These "free" electrons behave approximately like the particles in a one-dimensional box. 2

3 To understand the electronic structures of the three compounds you will study in this experiment, begin by considering the bonding between the carbon atoms in a very simple organic molecule, ethene (C 2 H 4 ). A line-bond structure for ethene is shown above, together with a more detailed orbital cartoon. Each carbon atom utilizes sp 2 hybrid orbitals to overlap with the 1s orbitals from two hydrogen atoms, while the remaining sp 2 hybrid orbital overlaps with an sp 2 hybrid orbital from the adjacent carbon atom. This "end-on" overlap is called σ-bonding ( sigma bonding ). This experiment is particularly concerned with a second type of bonding in which the p-orbitals on adjacent carbon atoms overlap. For purposes of electron bookkeeping, each p- orbital can be assumed to contain one electron. The overlap between p-orbitals on the adjacent carbons in ethene is "side-on" and termed a π-bond ( pi bond ). Hence, the double bond between the ethene carbon atoms actually consists of two distinct components, a σ-bond and a π- bond. More complex organic molecules containing alternating single and double bonds are said to be conjugated. The compounds used in this experiment are indeed conjugated, but the chemical structure of butadiene (shown below) is a useful, simpler example of a conjugated molecule. An examination of the π-bonding in butadiene shows that a p-orbital in each carbon atom overlaps with the p-orbital(s) of its neighbor(s). Again, each p-orbital is formally said to contain 1 electron. More extensively conjugated molecules called polyenes have formal structures with alternating single and double bonds along the carbon atom chain. The electrons in the π-bonds of polyenes can be considered to be delocalized over all the atoms of the conjugated chain, and thus can be thought of as moving somewhat freely along the length of the chain. Viewing each electron as a particle in a box is then a fairly crude, but physically reasonable, model of electrons moving along a chain of carbon atoms. For many compounds, it is remarkably successful in modeling the behavior of the quantum mechanical particle in a box. In this experiment, you will measure the light absorption properties of a carefully chosen set of organic molecules and relate the absorption spectra to the particle in a box quantum mechanical model for the electrons. 3

4 Pre-lab Review the instructions for the operation of the OceanOptics Spectrophotometer. You should have used this instrument in a previous lab and it will be used during this and future laboratory exercises. Thus, the operation of this instrument should be familiar to you. Please answer the following questions in your lab notebook. This assignment is due at the beginning of lab. You will not be allowed to start the experiment until this assignment has been completed and submitted to your TA. 1) The particle in the box energy, E n, involves Planck s constant, h, and the particle mass, m. What particle is involved in the transitions you will measure in this experiment? What are the values of h and m? 2) According to the assumptions of the particle-in-a-box experiment, how many freely moving π-electrons are there between the phenyl rings in each of the three organic compounds you will be studying? 3) Using the particle in the box equation and a wavelength of 400 nm, what is the length of the box of 1,4-diphenyl-1,3-butadiene? 4) The particle in a box theory assumes that potential energy is constant along the entire conjugated carbon-carbon chain. Are there any flaws in this assumption? Procedure Safety: Goggles must be worn at all times. Organic solvents (e.g., cyclohexane) should be collected in a separate container as waste. Do not uncap or empty the contents of the cuvettes at any point. Part 1 - Measuring the Spectra for Electrons in Boxes In this experiment, you will carry out absorbance measurements on three conjugated dyes for which the particle-in-a-box theory works very well. The compounds are 1,4-diphenyl-1,3- butadiene; 1,6-diphenyl-1,3,5-hexatriene; and 1,8-diphenyl-1,3,5,7-octatetraene. The chemical structures are: 4

5 Obtain the three pre-filled cuvettes corresponding to each of the following: 1,4-diphenyl-1,3- butadiene; 1,6-diphenyl-1,3,5-hexatriene; and 1,8-diphenyl-1,3,5,7-octatetraene. Each compound has been dissolved in cyclohexane and placed in a capped cuvette which is labeled accordingly with "1,4" "1,6" or "1,8." Please do not uncap or discard the solutions in the prefilled cuvettes. You will use the Ocean Optics spectrophotometer to acquire absorption spectra of these compounds. Be sure to calibrate the instrument. What will you use as a calibration blank? After calibration, adjust the wavelength (x-axis) display range to 250 nm nm. We are presently not interested in measuring the absorbance above 450 nm. As you acquire and save each spectrum, be sure to record in your notebook, along with a sketch of the spectrum, the lowest-energy local maximum in the absorbance, or λ max, to help you interpret the spectral data. Part 2 - Interpreting the Data Convert the wavelength associated with the lowest energy electronic transition to energy (in J). You will need to know specific values for E when you apply the energy expression given above. To calculate the length L of your experimental "box" using the eigenvalue expression, m is taken to be the mass of an electron and n i (n f ) is the initial (final) quantum level for the electronic transition. The value for n is determined by first counting the number of π electrons associated with the carbon bonds between the phenyl rings and then constructing an energy diagram in which energy levels are filled with electron pairs. For example, if a molecule has eight π electrons, the electronic configuration can be summarized by the following diagram. 5

6 After filling the lowest quantum levels, it becomes obvious that the lowest energy electronic transition for this particular system involves excitation of an electron from level n i = 4 to level n f = 5. Using the eigenvalue expression, we find DE molecule = E n,f - E n,i DE molecule = E 5 - E 4 = ( )h 2 8mL 2 where n f and n i are the final and initial quantum levels associated with the electronic transition. By substituting an experimental value for ΔE molecule, it is possible to solve this equation for L, the length of your one-dimensional experimental box. In this experiment, the box length is taken to the distance between the phenyl rings, as the phenyl rings represent the walls of the box. If we assume that the relevant carbon-carbon bonds in conjugated organic molecule are each nm long, we can calculate the theoretical box length for each molecule. You will want to compare each of your experimental box lengths to the theoretical values so that you can obtain a rough estimate of experimental error (i.e, % error). If the only energy changes accompanying light absorption were strictly electronic in nature, we would expect absorption spectra to show sharp maxima at the predicted wavelengths. In reality, such "line spectra" are generally observed only for isolated (e.g., gaseous) atoms, whereas substances in the liquid phase show broad absorption bands. Since the experiment is done under conditions that do not yield sharp spectral lines, use the lowest-energy (i.e., longest wavelength) local maximum, or λ max, in your calculations. Question Sheet: You must complete and turn in the question sheet provided to you by your teaching assistant before you leave the laboratory. 6

7 Glossary quantum number electronic state an index for labeling the different quantum a quantum-mechanical state of an atom or molecule mechanical states, e.g., electronic states of a associated with the arrangement of the electrons molecule, which is usually integer (0, 1, 2,...) or around the nuclei; each electronic state has a half-integer (1/2, 3/2, 5/2,...) corresponding electronic energy; electronic states are to be distinguished from vibrational or rotational states of molecules excited state any state of a quantum mechanical system such as a molecule which has higher energy than the ground state; there are many excited states, the lowest in energy is called the first excited state, the next lowest the second excited state, etc. ground state the lowest energy state of a quantum mechanical system, e.g., a molecule organic molecule a molecule containing carbon; typically only used to describe molecules that do not involve elements other than hydrogen and oxygen in addition to carbon; other molecules are called inorganic molecules quantum mechanics the theory of matter in which objects are described probabilistically in a way that combines both particle- and wave-like characteristics; the description exhibits dramatic differences from classical (Newtonian) mechanics for objects which are small in size and/or mass 7